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**From clusters of particles to 2D bubble clusters**

Edwin Flikkema, Simon Cox IMAPS, Aberystwyth University, UK

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**Introduction and overview**

The minimal perimeter problem for 2D equal area bubble clusters. Systems of interacting particles Global optimisation 2D particle clusters to 2D bubble clusters Voronoi construction 2D particle systems: -log(r) or 1/rp repulsive potential Harmonic or polygonal confining potentials Results

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**2D bubble clusters Minimal perimeter problem: 2D cluster of N bubbles.**

All bubbles have equal area. Free or confined to the interior of a circle or polygon. Minimize total perimeter (internal + external). Objective: apply techniques used in interacting particle clusters to this minimal perimeter problem.

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**Systems of interacting particles**

System energy: Usually: Example: Lennard-Jones potential: LJ13: Ar13 Used in: Molecular Dynamics, Monte Carlo, Energy landscapes

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**Energy landscapes Energy vs coordinate**

Stationary points of U: zero net force on each particle Minima of U correspond to (meta-)stable states. Global minimum is the most stable state. Local optimisation (finding a nearby minimum) relatively easy: Steepest descent, L-BFGS, Powell, etc. Global optimisation: hard. Energy vs coordinate Local optimisation

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**Global optimisation methods**

Inspired by simulated annealing: Basin hopping Minima hopping Evolutionary algorithms: Genetic algorithm Other: Covariance matrix adaption Simply starting from many random geometries

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**2D particle systems Energy: Repulsive inter-particle potential:**

Confining potential: or harmonic polygonal

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2D particle clusters Pictures of particle clusters: e.g. N=41, bottom 3 in energy

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**Particles to bubbles Qhull Surface Evolver particle cluster**

Voronoi cells optimized perimeter

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**2D particle clusters Polygonal confining potential: e.g. triangular**

unit vectors contour lines discontinuous gradient: smoothing needed?

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**Technical details List of unique 2D geometries produced**

Problem: permutational isomers. Distinguishing by energy U not sufficient: Spectrum of inter-particle distances compared. Gradient-based local optimisers have difficulty with polygonal potential due to discontinuous gradient Smoothing needed? Use gradient-less optimisers (e.g. Powell)?

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**Results: bubble clusters: Free, circle, hexagon**

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**Results: bubble clusters: pentagon, square, triangle**

Elec. J. Combinatorics 17:R45 (2010)

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Conclusions Optimal geometries of clusters of interacting particles can be used as candidates for the minimal perimeter problem. Various potentials have been tried. 1/r seems to work slightly better than –log(r). Using multiple potentials is recommended. Polygonal potentials have been introduced to represent confinement to a polygon

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Acknowledgements Simon Cox Adil Mughal

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**Energy landscapes Energy vs coordinate**

Stationary points of U: zero net force on each particle Minima of U correspond to (meta-)stable states. Global minimum is the most stable state. Saddle points (first order): transition states Network of minima connected by transition states Local optimisation (finding a nearby minimum) relatively easy: L-BFGS, Powell, etc. Global optimisation: hard. Local optimisation Energy vs coordinate

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2D clusters: perimeter is fit to data for free clusters

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